\subsection{Tony's comment}
He comment about the varying T approach: it is like to channel all magnetic-field-varying only affects two-body quantities T-matrix, not extra many-body effects. He actually suspected that there are important many-body effects because there are probably important effects when energy comparable to Fermi energy while in two-body the important effects only show up when it comparable to two-body energy such as level difference.  

He suggest me to calculate at this particular point, then varies the magnetic filed /$\eta$ and see how about the two-body changes and many-body changes.  

For the two-body density matrix of Richardson solution, he suggested me to look into the old 1957-early 60s' papers for something like ``asymptotic Bardeen Cooper Schrieffer''.  